How do I solve x^2 + x - 6 > 0 ?

This thing we have to solve is an inequality and the solution we are looking for is an entire range of real number, something like "every x between 1 and 2", for example. To do this we need to build a sign diagram and to build a sign diagram we need to find the root of the respective equation first. This is because the roots are when the right hand side (rhs) changes sign. In the intervals between two different roots the sign stay the same. In this case, the roots are x = -3 and x = 2. To compute the sign of the rhs before -3, we can simply compute the results when we substitute any (really, any!) number lower then -3, such as -4. We have: (-4)^2 -4 -6 = 6>0. So, the sign is positive. Same procedure for the (-3,2) interval and (2,infinity). The former gives us a negative sign, the latter a positive one. We want the intervals when the rhs is positive. Therefore the solution is: x<-3 and x>2.